I have found numerous references to some number theory that seems to draw a parallel to my whole computing sum's scenerio that has been driving me crazy as of recent.

The key term is digital root. It seems that my reducing the values of a number to a single digit by adding together each of the digits in the number is computing the digital root. Okay, no biggi there. There also seems to be a semi-weak relation to sigma codes, though I havn't quite decided yet if what I have run across is infact some property of sigma codes or not.

Originally I had used a double itterative loop to compute the digital root, but I have run across a much faster way of computing it for base10 (the 9 can be replaced with baseN - 1):

dr = 1 + (((n ** x) - 1) % 9)

And I have run across a curious conjecture in my recent insanity into this whole thing:

dr(n ** x) = dr(n ** dr(x))

It also seems that: if dr(n) == dr(i) then dr(n ** x) = dr(i ** x)

I am still trying to piece together a proof for this with a friend at work. As soon as we come up with one (more likely he will do all the work here, I am horrible at proofs), we have decided to see about finding a proof that describes the reproducable patterns that occure. And then it is off to explain why it is that predictable powers in the pattern intersect at a digital root of 1 at the same time, and a seperate predictable set of powers never reduce 1, and why it is that these powers are always a fixed distance apart.